MortalityLaw {MortalityLaws}

Fit Mortality Laws

Description

Fit parametric mortality models given a set of input data which can be represented by death counts and mid-interval population estimates (Dx, Ex) or age-specific death rates (mx) or death probabilities (qx). Using the argument law one can specify the model to be fitted. So far more than 27 parametric models have been implemented; check the availableLaws function to learn about the available options. The models can be fitted under the maximum likelihood methodology or by selecting a loss function to be optimised. See the implemented loss function by running the availableLF function.

Usage

MortalityLaw(x, Dx = NULL, Ex = NULL, mx = NULL, qx = NULL,
                law = NULL,
                opt.method = "LF2",
                parS = NULL,
                fit.this.x = x,
                custom.law = NULL,
                show = FALSE, ...)

Arguments

x

Vector of ages at the beginning of the age interval.

Dx

Object containing death counts. An element of the Dx object represents the number of deaths during the year to persons aged x to x+n.

Ex

Exposure in the period. Ex can be approximated by the mid-year population aged x to x+n.

mx

Life table death rate in age interval [x, x+n).

qx

Probability of dying in age interval [x, x+n).

law

The name of the mortality law/model to be used. e.g. gompertz, makeham, ... To investigate all the possible options, see availableLaws function.

opt.method

How would you like to find the parameters? Specify the function to be optimize. Available options: the Poisson likelihood function poissonL; the Binomial likelihood function -binomialL; and 6 other loss functions. For more details, check the availableLF function.

parS

Starting parameters used in the optimization process (optional).

fit.this.x

Select the ages to be considered in model fitting. By default fit.this.x = x. One may want to exclude from the fitting procedure, say, the advanced ages where the data is sparse.

custom.law

Allows you to fit a model that is not defined in the package. Accepts as input a function.

show

Choose whether to display a progress bar during the fitting process. Logical. Default: FALSE.

...

Arguments to be passed to or from other methods.

Details

Depending on the complexity of the model, one of following optimization strategies is employed:

1. Nelder-Mead method: approximates a local optimum of a problem with n variables when the objective function varies smoothly and is unimodal. For details see optim;

2. PORT routines: provides unconstrained optimization and optimization subject to box constraints for complicated functions. For details check nlminb;

3. Levenberg-Marquardt algorithm: damped least-squares method. For details check nls.lm.

Value

The output is of the "MortalityLaw" class with the components:

input

List with arguments provided in input. Saved for convenience.

info

Brief information about the model.

coefficients

Estimated coefficients.

fitted.values

Fitted values of the selected model.

residuals

Deviance residuals.

goodness.of.fit

List containing goodness of fit measures like AIC, BIC and log-Likelihood.

opt.diagnosis

Resultant optimization object useful for checking the convergence etc.

Author(s)

Marius D. Pascariu

See Also

availableLaws availableLF LifeTable ReadHMD

Examples

# Example 1: --------------------------
# Fit Makeham Model for Year of 1950.

x  <- 45:75
Dx <- ahmd$Dx[paste(x), "1950"]
Ex <- ahmd$Ex[paste(x), "1950"]

M1 <- MortalityLaw(x = x, Dx = Dx, Ex = Ex, law = 'makeham')

M1
ls(M1)
coef(M1)
summary(M1)
fitted(M1)
predict(M1, x = 45:95)
plot(M1)


# Example 2: --------------------------
# We can fit the same model using a different data format
# and a different optimization method.
x  <- 45:75
mx <- ahmd$mx[paste(x), ]
M2 <- MortalityLaw(x = x, mx = mx, law = 'makeham', opt.method = 'LF1')
M2
fitted(M2)
predict(M2, x = 55:90)

# Example 3: --------------------------
# Now let's fit a mortality law that is not defined
# in the package, say a reparameterized Gompertz in
# terms of modal age at death
# hx = b*exp(b*(x-m)) (here b and m are the parameters to be estimated)

# A function with 'x' and 'par' as input has to be defined, which returns
# at least an object called 'hx' (hazard rate).
my_gompertz <- function(x, par = c(b = 0.13, M = 45)){
  hx  <- with(as.list(par), b*exp(b*(x - M)) )
  return(as.list(environment()))
}

M3 <- MortalityLaw(x = x, Dx = Dx, Ex = Ex, custom.law = my_gompertz)
summary(M3)
# predict M3 for different ages
predict(M3, x = 85:130)


# Example 4: --------------------------
# Fit Heligman-Pollard model for a single
# year in the dataset between age 0 and 100 and build a life table.

x  <- 0:100
mx <- ahmd$mx[paste(x), "1950"] # select data
M4 <- MortalityLaw(x = x, mx = mx, law = 'HP', opt.method = 'LF2')
M4
plot(M4)

LifeTable(x = x, qx = fitted(M4))